90 research outputs found
-Penalization in Functional Linear Regression with Subgaussian Design
We study functional regression with random subgaussian design and real-valued
response. The focus is on the problems in which the regression function can be
well approximated by a functional linear model with the slope function being
"sparse" in the sense that it can be represented as a sum of a small number of
well separated "spikes". This can be viewed as an extension of now classical
sparse estimation problems to the case of infinite dictionaries. We study an
estimator of the regression function based on penalized empirical risk
minimization with quadratic loss and the complexity penalty defined in terms of
-norm (a continuous version of LASSO). The main goal is to introduce
several important parameters characterizing sparsity in this class of problems
and to prove sharp oracle inequalities showing how the -error of the
continuous LASSO estimator depends on the underlying sparsity of the problem
Greedy Algorithms for Cone Constrained Optimization with Convergence Guarantees
Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe
(FW) algorithms regained popularity in recent years due to their simplicity,
effectiveness and theoretical guarantees. MP and FW address optimization over
the linear span and the convex hull of a set of atoms, respectively. In this
paper, we consider the intermediate case of optimization over the convex cone,
parametrized as the conic hull of a generic atom set, leading to the first
principled definitions of non-negative MP algorithms for which we give explicit
convergence rates and demonstrate excellent empirical performance. In
particular, we derive sublinear () convergence on general
smooth and convex objectives, and linear convergence () on
strongly convex objectives, in both cases for general sets of atoms.
Furthermore, we establish a clear correspondence of our algorithms to known
algorithms from the MP and FW literature. Our novel algorithms and analyses
target general atom sets and general objective functions, and hence are
directly applicable to a large variety of learning settings.Comment: NIPS 201
Estimation in high dimensions: a geometric perspective
This tutorial provides an exposition of a flexible geometric framework for
high dimensional estimation problems with constraints. The tutorial develops
geometric intuition about high dimensional sets, justifies it with some results
of asymptotic convex geometry, and demonstrates connections between geometric
results and estimation problems. The theory is illustrated with applications to
sparse recovery, matrix completion, quantization, linear and logistic
regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change
Sparse recovery in convex hulls via entropy penalization
Let be a random couple in with unknown distribution
and be i.i.d. copies of Denote the
empirical distribution of Let be a dictionary that consists of functions. For denote Let
be a given loss function and
suppose it is convex with respect to the second variable. Let Finally, let be the
simplex of all probability distributions on Consider the
following penalized empirical risk minimization problem
\begin{eqnarray*}\hat{\lambda}^{\varepsilon}:={\mathop {argmin}_{\lambda\in
\Lambda}}\Biggl[P_n(\ell \bullet f_{\lambda})+\varepsilon
\sum_{j=1}^N\lambda_j\log \lambda_j\Biggr]\end{eqnarray*} along with its
distribution dependent version \begin{eqnarray*}\lambda^{\varepsilon}:={\mathop
{argmin}_{\lambda\in \Lambda}}\Biggl[P(\ell \bullet f_{\lambda})+\varepsilon
\sum_{j=1}^N\lambda_j\log \lambda_j\Biggr],\end{eqnarray*} where
is a regularization parameter. It is proved that the
``approximate sparsity'' of implies the ``approximate
sparsity'' of and the impact of ``sparsity'' on
bounding the excess risk of the empirical solution is explored. Similar results
are also discussed in the case of entropy penalized density estimation.Comment: Published in at http://dx.doi.org/10.1214/08-AOS621 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
SISSO: a compressed-sensing method for identifying the best low-dimensional descriptor in an immensity of offered candidates
The lack of reliable methods for identifying descriptors - the sets of
parameters capturing the underlying mechanisms of a materials property - is one
of the key factors hindering efficient materials development. Here, we propose
a systematic approach for discovering descriptors for materials properties,
within the framework of compressed-sensing based dimensionality reduction.
SISSO (sure independence screening and sparsifying operator) tackles immense
and correlated features spaces, and converges to the optimal solution from a
combination of features relevant to the materials' property of interest. In
addition, SISSO gives stable results also with small training sets. The
methodology is benchmarked with the quantitative prediction of the ground-state
enthalpies of octet binary materials (using ab initio data) and applied to the
showcase example of predicting the metal/insulator classification of binaries
(with experimental data). Accurate, predictive models are found in both cases.
For the metal-insulator classification model, the predictive capability are
tested beyond the training data: It rediscovers the available pressure-induced
insulator->metal transitions and it allows for the prediction of yet unknown
transition candidates, ripe for experimental validation. As a step forward with
respect to previous model-identification methods, SISSO can become an effective
tool for automatic materials development.Comment: 11 pages, 5 figures, in press in Phys. Rev. Material
Non-Negative Sparse Regression and Column Subset Selection with L1 Error
We consider the problems of sparse regression and column subset selection under L1 error. For both problems, we show that in the non-negative setting it is possible to obtain tight and efficient approximations, without any additional structural assumptions (such as restricted isometry, incoherence, expansion, etc.). For sparse regression, given a matrix A and a vector b with non-negative entries, we give an efficient algorithm to output a vector x of sparsity O(k), for which |Ax - b|_1 is comparable to the smallest error possible using non-negative k-sparse x. We then use this technique to obtain our main result: an efficient algorithm for column subset selection under L1 error for non-negative matrices
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